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Theorem oppgmnd 14827
Description: The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
Hypothesis
Ref Expression
oppgbas.1  |-  O  =  (oppg
`  R )
Assertion
Ref Expression
oppgmnd  |-  ( R  e.  Mnd  ->  O  e.  Mnd )

Proof of Theorem oppgmnd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppgbas.1 . . . 4  |-  O  =  (oppg
`  R )
2 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2oppgbas 14824 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 10 . 2  |-  ( R  e.  Mnd  ->  ( Base `  R )  =  ( Base `  O
) )
5 eqidd 2284 . 2  |-  ( R  e.  Mnd  ->  ( +g  `  O )  =  ( +g  `  O
) )
6 eqid 2283 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
7 eqid 2283 . . . 4  |-  ( +g  `  O )  =  ( +g  `  O )
86, 1, 7oppgplus 14822 . . 3  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  R ) x )
92, 6mndcl 14372 . . . 4  |-  ( ( R  e.  Mnd  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
1093com23 1157 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
118, 10syl5eqel 2367 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  O
) y )  e.  ( Base `  R
) )
12 simpl 443 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  R  e.  Mnd )
13 simpr3 963 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
z  e.  ( Base `  R ) )
14 simpr2 962 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
y  e.  ( Base `  R ) )
15 simpr1 961 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  x  e.  ( Base `  R ) )
162, 6mndass 14373 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( z  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R ) ) )  ->  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) ) )
1712, 13, 14, 15, 16syl13anc 1184 . . . 4  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( z ( +g  `  R ) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R ) ( y ( +g  `  R ) x ) ) )
1817eqcomd 2288 . . 3  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( z ( +g  `  R ) ( y ( +g  `  R
) x ) )  =  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x ) )
198oveq1i 5868 . . . 4  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( ( y ( +g  `  R ) x ) ( +g  `  O
) z )
206, 1, 7oppgplus 14822 . . . 4  |-  ( ( y ( +g  `  R
) x ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
2119, 20eqtri 2303 . . 3  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
226, 1, 7oppgplus 14822 . . . . 5  |-  ( y ( +g  `  O
) z )  =  ( z ( +g  `  R ) y )
2322oveq2i 5869 . . . 4  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( x ( +g  `  O ) ( z ( +g  `  R
) y ) )
246, 1, 7oppgplus 14822 . . . 4  |-  ( x ( +g  `  O
) ( z ( +g  `  R ) y ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2523, 24eqtri 2303 . . 3  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2618, 21, 253eqtr4g 2340 . 2  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x ( +g  `  O ) y ) ( +g  `  O ) z )  =  ( x ( +g  `  O ) ( y ( +g  `  O ) z ) ) )
27 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
282, 27mndidcl 14391 . 2  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  ( Base `  R
) )
296, 1, 7oppgplus 14822 . . 3  |-  ( ( 0g `  R ) ( +g  `  O
) x )  =  ( x ( +g  `  R ) ( 0g
`  R ) )
302, 6, 27mndrid 14394 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  R ) ( 0g
`  R ) )  =  x )
3129, 30syl5eq 2327 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  O ) x )  =  x )
326, 1, 7oppgplus 14822 . . 3  |-  ( x ( +g  `  O
) ( 0g `  R ) )  =  ( ( 0g `  R ) ( +g  `  R ) x )
332, 6, 27mndlid 14393 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  R ) x )  =  x )
3432, 33syl5eq 2327 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  O ) ( 0g
`  R ) )  =  x )
354, 5, 11, 26, 28, 31, 34ismndd 14396 1  |-  ( R  e.  Mnd  ->  O  e.  Mnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361  oppgcoppg 14818
This theorem is referenced by:  oppgmndb  14828  oppggrp  14830  gsumwrev  14839  gsumzoppg  15216  gsumzinv  15217  oppgtmd  17780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mnd 14367  df-oppg 14819
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